[Tekmen]

What could be the ring's power source?

There's one simliar technology called the "Inductive charging".
http://en.wikipedia.org/wiki/Inductive_charging

Quote:

Inductive charging uses an electromagnetic field to transfer energy between two objects. This is usually done with a charging station. Energy is sent through inductive coupling to an electrical device, which then can use that energy to charge batteries or run the device.

The wireless powering from an electronic toy might not be enough to activate your phone, but maybe just fine to light up a bulb on a ring.
It is UNBELIEVABLY AMAZING that Bandai manage to apply this tech to a children's toyline.